3.11 \(\int x^2 \tanh ^{-1}(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}) \, dx\)

Optimal. Leaf size=68 \[ -\frac{\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac{d \sqrt{d+e x^2}}{3 e^{3/2}}+\frac{1}{3} x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]

[Out]

(d*Sqrt[d + e*x^2])/(3*e^(3/2)) - (d + e*x^2)^(3/2)/(9*e^(3/2)) + (x^3*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/3

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Rubi [A]  time = 0.0347721, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6221, 266, 43} \[ -\frac{\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac{d \sqrt{d+e x^2}}{3 e^{3/2}}+\frac{1}{3} x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]],x]

[Out]

(d*Sqrt[d + e*x^2])/(3*e^(3/2)) - (d + e*x^2)^(3/2)/(9*e^(3/2)) + (x^3*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/3

Rule 6221

Int[ArcTanh[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*ArcT
anh[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /;
FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{1}{3} x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{3} \sqrt{e} \int \frac{x^3}{\sqrt{d+e x^2}} \, dx\\ &=\frac{1}{3} x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{6} \sqrt{e} \operatorname{Subst}\left (\int \frac{x}{\sqrt{d+e x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{6} \sqrt{e} \operatorname{Subst}\left (\int \left (-\frac{d}{e \sqrt{d+e x}}+\frac{\sqrt{d+e x}}{e}\right ) \, dx,x,x^2\right )\\ &=\frac{d \sqrt{d+e x^2}}{3 e^{3/2}}-\frac{\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac{1}{3} x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0465507, size = 56, normalized size = 0.82 \[ \frac{1}{9} \left (\frac{\left (2 d-e x^2\right ) \sqrt{d+e x^2}}{e^{3/2}}+3 x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]],x]

[Out]

(((2*d - e*x^2)*Sqrt[d + e*x^2])/e^(3/2) + 3*x^3*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/9

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Maple [B]  time = 0.03, size = 128, normalized size = 1.9 \begin{align*}{\frac{{x}^{3}}{3}{\it Artanh} \left ({x\sqrt{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{1}{3\,d}{e}^{{\frac{3}{2}}} \left ({\frac{{x}^{4}}{5\,e}\sqrt{e{x}^{2}+d}}-{\frac{4\,d}{5\,e} \left ({\frac{{x}^{2}}{3\,e}\sqrt{e{x}^{2}+d}}-{\frac{2\,d}{3\,{e}^{2}}\sqrt{e{x}^{2}+d}} \right ) } \right ) }-{\frac{1}{3\,d}\sqrt{e} \left ({\frac{{x}^{2}}{5\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{2\,d}{15\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2)),x)

[Out]

1/3*x^3*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))+1/3*e^(3/2)/d*(1/5*x^4/e*(e*x^2+d)^(1/2)-4/5*d/e*(1/3*x^2/e*(e*x^2+
d)^(1/2)-2/3*d/e^2*(e*x^2+d)^(1/2)))-1/3*e^(1/2)/d*(1/5*x^2*(e*x^2+d)^(3/2)/e-2/15*d/e^2*(e*x^2+d)^(3/2))

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Maxima [A]  time = 0.976157, size = 134, normalized size = 1.97 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{artanh}\left (\frac{\sqrt{e} x}{\sqrt{e x^{2} + d}}\right ) - \frac{3 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d}{45 \, d e^{\frac{3}{2}}} + \frac{3 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x^{2} + d} d^{2}}{45 \, d e^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="maxima")

[Out]

1/3*x^3*arctanh(sqrt(e)*x/sqrt(e*x^2 + d)) - 1/45*(3*(e*x^2 + d)^(5/2) - 5*(e*x^2 + d)^(3/2)*d)/(d*e^(3/2)) +
1/45*(3*(e*x^2 + d)^(5/2) - 10*(e*x^2 + d)^(3/2)*d + 15*sqrt(e*x^2 + d)*d^2)/(d*e^(3/2))

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Fricas [A]  time = 2.11433, size = 155, normalized size = 2.28 \begin{align*} \frac{3 \, e^{2} x^{3} \log \left (\frac{2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x + d}{d}\right ) - 2 \, \sqrt{e x^{2} + d}{\left (e x^{2} - 2 \, d\right )} \sqrt{e}}{18 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="fricas")

[Out]

1/18*(3*e^2*x^3*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) - 2*sqrt(e*x^2 + d)*(e*x^2 - 2*d)*sqrt(e))/
e^2

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Sympy [A]  time = 8.62313, size = 65, normalized size = 0.96 \begin{align*} \begin{cases} \frac{2 d \sqrt{d + e x^{2}}}{9 e^{\frac{3}{2}}} + \frac{x^{3} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{3} - \frac{x^{2} \sqrt{d + e x^{2}}}{9 \sqrt{e}} & \text{for}\: e \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*atanh(x*e**(1/2)/(e*x**2+d)**(1/2)),x)

[Out]

Piecewise((2*d*sqrt(d + e*x**2)/(9*e**(3/2)) + x**3*atanh(sqrt(e)*x/sqrt(d + e*x**2))/3 - x**2*sqrt(d + e*x**2
)/(9*sqrt(e)), Ne(e, 0)), (0, True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError